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\begin{table}[!tbp]
\caption{Simulation results: Quadratic outcome models\label{tb_Quadratic_2}} 
{\centering
\begin{tabular}{lrrcrcrrcrrcrrcrcrr}
\hline
\multicolumn{1}{l}{\ }&\multicolumn{2}{c}{\ $n = 200$}&\multicolumn{1}{c}{\ }&\multicolumn{1}{c}{\ }&\multicolumn{1}{c}{\ }&\multicolumn{2}{c}{\ $n = 1000$}&\multicolumn{1}{c}{\ }&\multicolumn{2}{c}{\ }&\multicolumn{1}{c}{\ }&\multicolumn{2}{c}{\ $n = 200$}&\multicolumn{1}{c}{\ }&\multicolumn{1}{c}{\ }&\multicolumn{1}{c}{\ }&\multicolumn{2}{c}{\ $n = 1000$}\tabularnewline
\cline{2-3} \cline{7-8} \cline{13-14} \cline{18-19}
\multicolumn{1}{l}{}&\multicolumn{1}{c}{Bias}&\multicolumn{1}{c}{RMSE}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{Bias}&\multicolumn{1}{c}{RMSE}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{Bias}&\multicolumn{1}{c}{RMSE}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{}&\multicolumn{1}{c}{Bias}&\multicolumn{1}{c}{RMSE}\tabularnewline
\hline
{\ \multicolumn{14}{l}{\textbf{Quadratic outcome model 1: correct PS model}}&&&&&&&&&&&&&&&&&&\tabularnewline
~~Unweighted&$-6.85$&$ 8.93$&&$$&&$-6.91$&$  7.36$&&$$&$$&&$ 7.16$&$ 9.77$&&$$&&$ 6.94$&$7.52$\tabularnewline
~~\textbf{nDBW DR}&$-2.16$&$ 6.08$&&$$&&$-0.48$&$  2.66$&&$$&$$&&$-0.86$&$ 5.67$&&$$&&$-0.20$&$2.53$\tabularnewline
~~MLE DR&$-1.43$&$ 7.45$&&$$&&$-0.31$&$  3.53$&&$$&$$&&$-0.53$&$ 6.95$&&$$&&$-0.14$&$2.96$\tabularnewline
~~CBPS DR&$-1.58$&$ 6.63$&&$$&&$-0.39$&$  3.11$&&$$&$$&&$-0.55$&$ 6.18$&&$$&&$-0.14$&$2.79$\tabularnewline
~~Calibrated weighting DR&$-1.78$&$ 6.15$&&$$&&$-0.45$&$  2.81$&&$$&$$&&$-0.64$&$ 5.74$&&$$&&$-0.17$&$2.56$\tabularnewline
~~Entropy balancing DR&$-3.68$&$ 6.86$&&$$&&$-2.72$&$  3.79$&&$$&$$&&$-2.13$&$ 5.92$&&$$&&$-1.67$&$2.94$\tabularnewline
~~True propensity score DR~~&$-1.63$&$ 7.58$&&$$&&$-0.30$&$  3.71$&&$$&$$&&$-0.78$&$ 6.88$&&$$&&$-0.22$&$3.15$\tabularnewline
\hline
{\ \multicolumn{14}{l}{\textbf{Quadratic outcome model 1: misspecified PS model}}&&&&&&&&&&&&&&&&&&\tabularnewline
~~Unweighted&$-6.95$&$ 9.03$&&$$&&$-6.85$&$  7.30$&&$$&$$&&$ 6.85$&$ 9.42$&&$$&&$ 6.99$&$7.55$\tabularnewline
~~\textbf{nDBW DR}&$-1.89$&$ 6.17$&&$$&&$-0.47$&$  2.66$&&$$&$$&&$ 0.32$&$ 6.24$&&$$&&$ 1.00$&$2.83$\tabularnewline
~~MLE DR&$ 1.71$&$25.10$&&$$&&$17.40$&$174.96$&&$$&$$&&$ 1.02$&$ 7.76$&&$$&&$ 1.34$&$3.51$\tabularnewline
~~CBPS DR/BRDR&$-2.04$&$ 6.71$&&$$&&$-0.09$&$  3.13$&&$$&$$&&$ 1.32$&$ 7.02$&&$$&&$ 1.21$&$3.21$\tabularnewline
~~Calibrated weighting DR&$-1.83$&$ 6.22$&&$$&&$-0.43$&$  2.73$&&$$&$$&&$ 0.65$&$ 6.30$&&$$&&$ 1.07$&$2.87$\tabularnewline
~~Entropy balancing DR&$-3.63$&$ 6.87$&&$$&&$-2.40$&$  3.59$&&$$&$$&&$-0.95$&$ 6.30$&&$$&&$-0.62$&$2.70$\tabularnewline
~~True propensity score DR~~&$-1.57$&$ 7.53$&&$$&&$-0.27$&$  3.68$&&$$&$$&&$-0.78$&$ 7.05$&&$$&&$-0.23$&$3.11$\tabularnewline
\hline
{\ \multicolumn{14}{l}{\textbf{Quadratic outcome model 2: correct PS model}}&&&&&&&&&&&&&&&&&&\tabularnewline
~~Unweighted&$ 6.75$&$ 9.34$&&$$&&$ 6.93$&$  7.52$&&$$&$$&&$-6.90$&$ 8.88$&&$$&&$-6.88$&$7.33$\tabularnewline
~~\textbf{nDBW DR}&$-1.60$&$ 6.20$&&$$&&$-0.39$&$  2.69$&&$$&$$&&$-3.03$&$ 6.61$&&$$&&$-0.71$&$2.86$\tabularnewline
~~MLE DR&$-1.11$&$ 7.63$&&$$&&$-0.12$&$  3.43$&&$$&$$&&$-1.53$&$ 9.63$&&$$&&$-0.17$&$4.80$\tabularnewline
~~CBPS DR&$-1.33$&$ 6.87$&&$$&&$-0.23$&$  3.16$&&$$&$$&&$-1.75$&$ 7.36$&&$$&&$-0.33$&$3.74$\tabularnewline
~~Calibrated weighting DR&$-1.50$&$ 6.38$&&$$&&$-0.32$&$  2.84$&&$$&$$&&$-2.10$&$ 6.51$&&$$&&$-0.49$&$3.06$\tabularnewline
~~Entropy balancing DR&$-4.30$&$ 7.49$&&$$&&$-3.42$&$  4.36$&&$$&$$&&$-5.09$&$ 7.81$&&$$&&$-4.02$&$4.86$\tabularnewline
~~True propensity score DR~~&$-1.26$&$ 7.52$&&$$&&$-0.18$&$  3.54$&&$$&$$&&$-1.87$&$10.22$&&$$&&$-0.31$&$5.16$\tabularnewline
\hline
{\ \multicolumn{14}{l}{\textbf{Quadratic outcome model 2: misspecified PS model}}&&&&&&&&&&&&&&&&&&\tabularnewline
~~Unweighted&$ 7.05$&$ 9.65$&&$$&&$ 6.94$&$  7.54$&&$$&$$&&$-7.00$&$ 9.00$&&$$&&$-6.85$&$7.34$\tabularnewline
~~\textbf{nDBW DR}&$ 4.31$&$ 8.24$&&$$&&$ 5.18$&$  6.03$&&$$&$$&&$-6.63$&$ 8.71$&&$$&&$-4.70$&$5.38$\tabularnewline
~~MLE DR&$10.72$&$31.50$&&$$&&$26.53$&$186.73$&&$$&$$&&$-7.23$&$10.21$&&$$&&$-6.63$&$7.37$\tabularnewline
~~CBPS DR/BRDR&$ 5.69$&$ 9.57$&&$$&&$ 6.97$&$  7.83$&&$$&$$&&$-7.15$&$ 9.54$&&$$&&$-7.01$&$7.61$\tabularnewline
~~Calibrated weighting DR&$ 4.67$&$ 8.46$&&$$&&$ 5.46$&$  6.30$&&$$&$$&&$-6.40$&$ 8.63$&&$$&&$-5.40$&$6.05$\tabularnewline
~~Entropy balancing DR&$ 2.30$&$ 7.29$&&$$&&$ 3.15$&$  4.41$&&$$&$$&&$-9.16$&$10.71$&&$$&&$-8.50$&$8.88$\tabularnewline
~~True propensity score DR~~&$-1.04$&$ 7.63$&&$$&&$-0.25$&$  3.47$&&$$&$$&&$-1.90$&$10.36$&&$$&&$-0.31$&$5.24$\tabularnewline
\hline
\end{tabular}}
\parbox{0.99\textwidth}
	{Notes: This simulation compares the performance of various methods 
	for estimating propensity scores and (inverse probability) weights 
	by investigating combinations of two versions of the true outcome model 
	(Quadratic~1 and 2)
	and two versions of coefficients for the propensity score model (type~A and B)
	with the two different numbers of observations ($n = 200$ and $n = 1000$).
	For each estimation method, I use two propensity score model specifications 
	(correct and misspecified) and report the bias and RMSE for each in the table.}\end{table}
